# On $L_2$-consistency of nearest neighbor matching

## TL;DR

This paper proves that nearest neighbor matching (NNM) is $L_2$-consistent in finite dimensions without requiring smoothness or boundedness, aiding statistical inference with biased samples.

## Contribution

It establishes the $L_2$-consistency of NNM under minimal assumptions, expanding understanding of its theoretical properties in biased sampling scenarios.

## Key findings

- NNM is $L_2$-consistent without smoothness or boundedness assumptions
- Discussion of applications and limitations of NNM
- Comparison of NNM with inverse probability weighting

## Abstract

Biased sampling and missing data complicates statistical problems ranging from causal inference to reinforcement learning. We often correct for biased sampling of summary statistics with matching methods and importance weighting. In this paper, we study nearest neighbor matching (NNM), which makes estimates of population quantities from biased samples by substituting unobserved variables with their nearest neighbors in the biased sample. We show that NNM is $L_2$-consistent in the absence of smoothness and boundedness assumptions in finite dimensions. We discuss applications of NNM, outline the barriers to generalizing this work to separable metric spaces, and compare this result to inverse probability weighting.